## A remark on integral geometry.(English)Zbl 1021.53049

Ghys, Étienne (ed.) et al., Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 1. Genève: L’Enseignement Mathématique. Monogr. Enseign. Math. 38, 59-83 (2001).
Let $$f=\left\{ f_{1},...,f_{n}\right\}$$ be $$n$$ homogeneous polynomials in $$(n+1)$$ variables and let $$\chi(f)$$ denote the cardinality of the simultaneous solutions of the equations $$f_{i}=0$$, for $$i=1,...,n$$, over the reals. In [Computational Algebraic Geometry, Nice 1992, Progr. Math. 109, 267-285 (1993; Zbl 0851.65031)] M. Shub and S. Smale proved that the “average” value of $$\chi(f)$$, as $$f_{i}$$ varies over the projective space of homogeneous polynomials of degree $$d_{i}$$, is equal to $$\sqrt{d_{1}...d_{n}}$$.
In this nice paper the authors show how the previous Bezout’s theorem can be derived from a universal integral formula, by introducing a “lifting principle” which allows them to use the push forward of the de Rham theory.
For the entire collection see [Zbl 0988.00114].

### MSC:

 53C65 Integral geometry

### Keywords:

Bezout’s theorem; integral geometry

Zbl 0851.65031